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Tagged with integral or integration
1,506 questions
3
votes
1
answer
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Approximating a multiple sum with an integral
Hi,
I want to approximate a multiple sum of the form
$$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$
where each $x_i$ is an integer between $0$ and $n$,
by an integral
$$\int_{x_1+x_2+\...
- 491
32
votes
4
answers
18k
views
About the Riemann integrability of composite functions
When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions.
For the composite function $f \circ g$, He presented three cases:
1) ...
- 627
6
votes
0
answers
2k
views
Fourier transforms via Kurzweil-Henstock integral on locally compact commutative groups
Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?
- 4,351
9
votes
1
answer
2k
views
Kullback-Leibler divergence of scaled non-central Student's T distribution
What is the Kullback-Leibler divergence of two Student's T distributions that have been shifted and scaled? That is, $\textrm{D}_{\textrm{KL}}(k_aA + t_a; k_bB + t_b)$ where $A$ and $B$ are Student's ...
- 598
19
votes
5
answers
18k
views
Visualization of Riemann–Stieltjes Integrals
The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–...
- 5,935
8
votes
4
answers
2k
views
An integral that somehow equals pi^2/6 and involves dilogarithms?
I am attempting to show that
$$ \sum_{k \ge 1}^\infty {k^2 x^k \over (1+x^k)^2} \sim (1-x)^{-3} {\pi^2 \over 6} $$
as $x$ approaches 1 from below. The sum can be approximated by the integral
$$ \...
- 14k